Interesting idea. Here is the relevant section of the 14th Amendment:
"Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws."
I do not see how the bold text applies to abortion, unless the State is committing the abortion. The portion in italics recognizes rights of persons but does not define persons. Abortion supporters would argue a baby is not a person until it is born. (Of course, by that logic, they could also claim blacks are not persons.)
If the state refuses to protect the child, then it is effectively denying them of its right to life. The 14th Amendment has been used when writing Federal legislation intended to protect blacks against lynching and other crimes by private citizens against life, liberty, and property.